Elliptic Racah polynomials

TL;DR

This paper introduces elliptic Racah polynomials as a new generalization of classical Racah polynomials, derived from a finite discrete reduction of the difference Heun equation, with established recurrence and orthogonality relations.

Contribution

It presents the first elliptic generalization of Racah polynomials, connecting them to the difference Heun equation and recovering known $q$-Racah polynomials in a limit.

Findings

Derived elliptic Racah polynomials from the difference Heun equation

Established three-term recurrence and orthogonality relations

Recovered $q$-Racah polynomials as a limit case

Abstract

Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic Racah polynomials. The well-known $q$-Racah polynomials of Askey and Wilson are recovered as a trigonometric limit.